Partial Derivatives - Chain Rule / Polar Coordinates Problem

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(please see image link below) Currently stuck on this problem involving the chain rule and partial derivatives. Having looked at the highlighted part of the solution, I can't figure out two things:

  • Surely the $\sin\psi$ term should be zero, as it's just taking the partial derivative with respect to $r$ of $\sin\psi$?
  • Likewise, shouldn't the $r \cos\psi$ term be simply $\cos\psi$, as it's just the partial derivative of $\sin\psi$ with respect to $\psi$?

Any advice would be much appreciated. partial derivative chain rule problem